Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Publications of the Research Institute for Mathematical Sciences

A Product Formula and Its Application to the Schrödinger Equation

Takashi Ichinose

Published Aug 31, 1980• 14 citations
Publisher

Abstract

A product formula is given which represents the unitary group for the form sum of a pair of nonnegative selfadjoint operators in a Hilbert space.
DOI
10.2977/prims/1195187217

Related papers

Product formula algorithms to solve the time-dependent schrodinger equation: application to two-particle quantum interference

Hans A. De Raedt, Kristel F. L. Michielsen

Host Publication•Jan 1, 1995•0 citations

On a Product Formula for Quantum Zeno Dynamics

Takashi Ichinose

$t$ as a result, although it seems to need to assume that each measurement is an instantaneous ideal measurement. The decay of the system can be slowed down or even prevented. This is called quantum Zeno eect. The problem was discussed in Beskow-Nilsson [BN], Misra-Sudarshan [MS]. Zeno is a Greek philosopher who said something like “A flying arrow does not fly”. The aim of this talk is to present the following product formula on quantum Zeno dynamics. This note is mainly based on the joint work with P.Exner [EI]. Theorem. Let $H$ be a nonnegative selfadjoint operator in a separable Hilbert space $ \mathcal{H}$

•Jan 1, 2006•0 citations

Product formula algorithms for solving the time dependent Schrödinger equation

Hans A. De Raedt

Computer Physics Reports•Dec 1, 1987•224 citations

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group

M. Kibler

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU (2). A single formula for the bases is obtained from a polar decomposition of SU (2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

arXiv:0907.2838•Journal of Physics A Mathematical and Theoretical•Aug 12, 2009•58 citations

Perturbations of the Schroedinger equation by potentials with small support, semigroup by product formulas, and applications to quantum mechanics

Charles N. Friedman

University Microfilms International eBooks•Jan 1, 1971•1 citation

Two sample constructions of all the Hilbert spaces for a given operator $H$ of an observable

Miloslav Znojil

An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $\Theta$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $\lambda \in (-1,1)$ in matrix $H^{(N)}(\lambda)$, the complete $N-$parametric set of metrics $\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)$ is recurrently defined by closed formula.

•Jun 26, 2008•0 citations
ttt as a result, although it seems to need to assume that each measurement is an instantaneous ideal measurement. The decay of the system can be slowed down or even prevented. This is called quantum Zeno eect. The problem was discussed in Beskow-Nilsson [BN], Misra-Sudarshan [MS]. Zeno is a Greek philosopher who said something like “A flying arrow does not fly”. The aim of this talk is to present the following product formula on quantum Zeno dynamics. This note is mainly based on the joint work with P.Exner [EI]. Theorem. Let HHH be a nonnegative selfadjoint operator in a separable Hilbert space $ \mathcal{H}$
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU (2). A single formula for the bases is obtained from a polar decomposition of SU (2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
Two sample constructions of all the Hilbert spaces for a given operator HHH of an observable
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators Θ\ThetaΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)\lambda \in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(\lambda)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)\Theta^{(N)}_{\alpha_1,\alpha_2,...,\alpha_N}(\lambda)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.